The official Brooks saddle cover doesn't really cover the bottom of the saddle leather - I suppose they expect one to have 'mudguards' on the machine.
The official Selle Anatomica saddle cover wraps up around the seatpost, providing full coverage. It costs more.
I purchased a rain cover for my wife to use with her B17 and unlike my older Brooks rain cover, this one is like you describe the Selle one. ie it has much more material and cinches up and protects the bottom of the seat also. I'd have to check who makes it, can't recall as I bought it a bunch of years ago.